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THE controversial topic of Mengenkhre (theory of sets, or
classes, particularly of infinite sets) created in 1874-95 by
Georg Cantor (1845-1918) may well be taken, out of its chrono-
logical order, as the conclusion of the whole story. This topic
typifies for mathematics the general collapse of those principles
which the prescient seers of the nineteenth century, foreseeing
everything but the grand debacle, believed to be fundamentally
sound in all things from physical science to democratic govern-
If 'collapse' is perhaps too strong to describe the transition
the world is doing its best to enjoy, it is nevertheless true that
the evolution of scientific ideas is now proceeding so vertigi-
nously that evolution is barely distinguishable from revolution,
Without the errors of the past as a deep-seated focus of dis-
turbance the present upheaval in physical science would per-
haps not have happened; but to credit our predecessors with all
the inspiration for what our own generation is doing, is to give
them more than their due. This point is worth a moment's
consideration, as some may be tempted to say that the corre-
sponding devolution* in mathematical thinking, whose begin-
nings are now plainly apparent, is merely an echo of Zeno and
other doubters of ancient Greece,
The difficulties of Pythagoras over the square root of 2 and
the paradoxes of Zeno on continuity (or Infinite divisibility')
are - so far as we know - the origins of our present mathe-
matical schism. Mathematicians to-day who pay any attention
to the philosophy (or foundations) of their subject are split into
at least two factions, apparently beyond present hope of recon-
ciliation, over the validity of the reasoning used in mathemati-